MATH 3150: PDE FOR ENGINEERS
MIDTERM TEST #1
Name:
Work out everything as far as you can before making decimal approximations.
1. Let α be a constant. Find
(a)
d αx
e
dx
(b)
Z
eαx dx
Check your answer (by differentiating it to see if you get the eαx back
again). You will need to use the answer later on.
(c)
d αx
xe
dx
(d)
Z
xeαx dx
Hint: watch for α = 0.
Date: July 16, 2001.
1
2
2.
MATH 3150: PDE FOR ENGINEERS
MIDTERM TEST #1
√
f (x) = α 2eα(x−π)
−π ≤x<π
where α is a constant, and f (x) has 2π period.
(a) Draw the graph of f (x) assuming α = 1.
(b) Calculate the complex Fourier series of f (x). (The amplitudes depend
on α.) Hint: since α is just a constant, your integrals are products of
exponential functions. Use the rule eA eB = eA+B .
(c) Calculate the energy of f (x) (it depends on α). What happens as α → ∞?
Hint: the function gets very wild.
(d) Calculate the energy stored in each amplitude. Approximately how much
energy is stored in the amplitudes ck for k a positive or negative number
which is small (close to zero) compared with α when α is large? Hint: it
better turn out like Heisenberg’s uncertainty principle—a long list of nearly
constant amplitudes.
MATH 3150: PDE FOR ENGINEERS
3.
Calculate
Z
2π
3
esin
0
(hint: don’t work out any integrals).
x
MIDTERM TEST #1
3
− ecos
x
dx
3
4
MATH 3150: PDE FOR ENGINEERS
MIDTERM TEST #1
4.
(a) Find the real Fourier series of f (x) = cos2 x − sin2 x (hint: don’t calculate
any integrals).
(b) The same for f (x) = sin x cos x.
MATH 3150: PDE FOR ENGINEERS
5.
MIDTERM TEST #1
5
The ordinary differential equation
d2 f
df
+µ +f =0
dt2
dt
(where µ is a real constant) has a periodic solution f (x) which is not constant.
Plug in a complex Fourier series for f (x) to find the possible values of µ.
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6.
MATH 3150: PDE FOR ENGINEERS
MIDTERM TEST #1
Suppose that f (x) has period T . Let
g(x) = f (T x/S)
where S is a positive constant.
(a) g(x) is periodic. With what period?
(b) Calculate the energy of g(x), expressed in terms of the energy of f (x).
(c) Express the complex amplitudes of g(x) in terms of those of f (x).